Regular Talk: Toward combinatorial proof of P < NP. Basic approach

Abstract

We present a plausibe “school-algebraic” condition C0
that infers,
in Peano Arithmetic, the negative solution (abbr.: P < NP) to the familiar
open problem P ?= NP. C0 expresses that a slight
modification of the ordinary DNF conversion algorithm needs exponential size
inputs in order to produce a certain big and complex output. This output is
explicitly defined and its structure can be analyzed by standard methods of
asymptotic combinatorics in order to achieve a desired proof of
C0. C0 also admits purely
combinatorial tree-presentation. We believe that our approach might accelerate
fulfillment of Harvey Friedman’s prophecy: “2050, P != NP. Detailed
combinatorial work on easier problems, leading up to the full result”.